What is the radius for the circle given by the equation $x^2+(y-1)^2=12$?
Round your answer to the nearest thousandth.
Answer (1)
Identify the equation of the circle and relate it to the general form.
Determine that r 2 = 12 .
Calculate the radius r = 12 = 2 3 .
Approximate the radius to the nearest thousandth: 3.464 .
Explanation
Problem Analysis We are given the equation of a circle: x 2 + ( y − 1 ) 2 = 12 . Our goal is to find the radius of this circle and round it to the nearest thousandth.
Relate to the general equation The general equation of a circle with center ( h , k ) and radius r is given by ( x − h ) 2 + ( y − k ) 2 = r 2 . Comparing this with our given equation x 2 + ( y − 1 ) 2 = 12 , we can identify that r 2 = 12 .
Find the radius To find the radius r , we take the square root of both sides of the equation r 2 = 12 . This gives us r = 12 .
Simplify the radical We can simplify the radical as follows: 12 = 4 ⋅ 3 = 4 ⋅ 3 = 2 3 .
Approximate the value Now, we need to approximate the value of 2 3 to the nearest thousandth. 2 3 ≈ 3.464101615... . Rounding this to the nearest thousandth gives us 3.464 .
Final Answer Therefore, the radius of the circle is approximately 3.464 .
Examples
Circles are fundamental in many real-world applications. For instance, in engineering, circles are used in the design of gears, wheels, and pipes. Knowing the equation of a circle allows engineers to determine its radius, which is crucial for ensuring that these components fit together correctly and function as intended. Similarly, in architecture, circular shapes are often used for aesthetic and structural purposes, and understanding their properties is essential for creating stable and visually appealing designs.
